↑

Properties

Label 1-4011-4011.677-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.843 - 0.537i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.471 − 0.882i)2-s + (−0.556 + 0.831i)4-s + (0.716 − 0.697i)5-s + (0.995 + 0.0990i)8-s + (−0.952 − 0.303i)10-s + (0.998 − 0.0550i)11-s + (−0.490 + 0.871i)13-s + (−0.381 − 0.924i)16-s + (0.391 + 0.920i)17-s + (−0.952 + 0.303i)19-s + (0.180 + 0.983i)20-s + (−0.518 − 0.854i)22-s + (−0.00551 + 0.999i)23-s + (0.0275 − 0.999i)25-s + (0.999 + 0.0220i)26-s + ⋯
L(s)  = 1  + (−0.471 − 0.882i)2-s + (−0.556 + 0.831i)4-s + (0.716 − 0.697i)5-s + (0.995 + 0.0990i)8-s + (−0.952 − 0.303i)10-s + (0.998 − 0.0550i)11-s + (−0.490 + 0.871i)13-s + (−0.381 − 0.924i)16-s + (0.391 + 0.920i)17-s + (−0.952 + 0.303i)19-s + (0.180 + 0.983i)20-s + (−0.518 − 0.854i)22-s + (−0.00551 + 0.999i)23-s + (0.0275 − 0.999i)25-s + (0.999 + 0.0220i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417835883 - 0.4134788991i\)
\(L(\frac12)\) \(\approx\) \(1.417835883 - 0.4134788991i\)
\(L(1)\) \(\approx\) \(0.9011098724 - 0.3310711389i\)
\(L(1)\) \(\approx\) \(0.9011098724 - 0.3310711389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.471 - 0.882i)T \)
5 \( 1 + (0.716 - 0.697i)T \)
11 \( 1 + (0.998 - 0.0550i)T \)
13 \( 1 + (-0.490 + 0.871i)T \)
17 \( 1 + (0.391 + 0.920i)T \)
19 \( 1 + (-0.952 + 0.303i)T \)
23 \( 1 + (-0.00551 + 0.999i)T \)
29 \( 1 + (0.724 - 0.689i)T \)
31 \( 1 + (-0.904 - 0.426i)T \)
37 \( 1 + (0.904 - 0.426i)T \)
41 \( 1 + (0.789 - 0.614i)T \)
43 \( 1 + (-0.148 + 0.988i)T \)
47 \( 1 + (-0.709 - 0.705i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (-0.982 + 0.186i)T \)
61 \( 1 + (-0.224 - 0.974i)T \)
67 \( 1 + (0.874 + 0.485i)T \)
71 \( 1 + (0.340 + 0.940i)T \)
73 \( 1 + (-0.988 + 0.153i)T \)
79 \( 1 + (-0.234 + 0.972i)T \)
83 \( 1 + (0.863 - 0.504i)T \)
89 \( 1 + (0.999 - 0.0440i)T \)
97 \( 1 + (0.973 - 0.229i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.30802251364494839136294352323, −17.84407226760959069871565107136, −17.187253311781662024280513692333, −16.6346909793402175301895420726, −15.903766135195985125500190090489, −14.96228203235871091754801030619, −14.54935230837041987271274252572, −14.14372426192184906680596297372, −13.20957503113152174957769273271, −12.57361250037335224772417731924, −11.50595594950355687298744007919, −10.61354937486690629124022904526, −10.248710986520977899491215393726, −9.33146179182641883666731516020, −8.97672387526670788375417564229, −7.93311167611584726746435923693, −7.30390923422550340101698971506, −6.44501251000132031103873313604, −6.23632269643025534015041290299, −5.138615740483782354344209091703, −4.632215280492956800329402693924, −3.44239924399957619761336115709, −2.56242715291905213590791572487, −1.62964808181049404288401690497, −0.62614445871512118414564675726, 0.83939162371957125409105586391, 1.76532889742240410512943404033, 2.06760869299925403699995047536, 3.28166216109275898541672894977, 4.18116723591455031133348381921, 4.55540304125297088849784339974, 5.729450079888776695278162076025, 6.37505978597761422846463741602, 7.39466166333733077241457930182, 8.23661059362374267215060643998, 8.85780884253264408559468158309, 9.59768418655052200648169023646, 9.83952508122304127066680767755, 10.900234672855484926356090953951, 11.49912198747236593308389962875, 12.32286756032397528131236526017, 12.71445353984425525871927165242, 13.48938053056717987969586755678, 14.21900278363611924967801355670, 14.75721692546793877768904060087, 16.02163254305397683772121636466, 16.727235306535580055806566497246, 17.18883231993919249750321546084, 17.56170729941522405917455889784, 18.54124331028447996064881235170

Graph of the $Z$-function along the critical line